VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.7% → 96.0%

Time: 36.9s

Alternatives: 7

Speedup: 3.4×

• could not determine a ground truth

  1. f = -1.311756926879144e+232

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t_0}\\ t_2 := e^{-t_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t_1 + t_2}{t_1 - t_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Initial Program: 6.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ t_1 := e^{t_0}\\ t_2 := e^{-t_0}\\ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{t_1 + t_2}{t_1 - t_2}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Alternative 1: 96.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\log f - \log \left(\frac{4}{\pi}\right)\right)}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (* 4.0 (- (log f) (log (/ 4.0 PI)))) PI))

C

double code(double f) {
	return (4.0 * (log(f) - log((4.0 / ((double) M_PI))))) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (4.0 * (Math.log(f) - Math.log((4.0 / Math.PI)))) / Math.PI;
}

Python

def code(f):
	return (4.0 * (math.log(f) - math.log((4.0 / math.pi)))) / math.pi

Julia

function code(f)
	return Float64(Float64(4.0 * Float64(log(f) - log(Float64(4.0 / pi)))) / pi)
end

MATLAB

function tmp = code(f)
	tmp = (4.0 * (log(f) - log((4.0 / pi)))) / pi;
end

Wolfram

code[f_] := N[(N[(4.0 * N[(N[Log[f], $MachinePrecision] - N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 5.8%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

2. Taylor expanded in f around 0 97.2%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi}} \]
3. Step-by-step derivation

   1. *-commutative 97.2%

      \[\leadsto -\color{blue}{\frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} \cdot 4} \]

   2. associate-*l/ 97.2%

      \[\leadsto -\color{blue}{\frac{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f\right) \cdot 4}{\pi}} \]

   3. mul-1-neg 97.2%

      \[\leadsto -\frac{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + \color{blue}{\left(-\log f\right)}\right) \cdot 4}{\pi} \]

   4. unsub-neg 97.2%

      \[\leadsto -\frac{\color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) - \log f\right)} \cdot 4}{\pi} \]

   5. distribute-rgt-out-- 97.2%

      \[\leadsto -\frac{\left(\log \left(\frac{2}{\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}}\right) - \log f\right) \cdot 4}{\pi} \]

   6. metadata-eval 97.2%

      \[\leadsto -\frac{\left(\log \left(\frac{2}{\pi \cdot \color{blue}{0.5}}\right) - \log f\right) \cdot 4}{\pi} \]

4. Simplified 97.2%

   \[\leadsto -\color{blue}{\frac{\left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right) \cdot 4}{\pi}} \]

5. Taylor expanded in f around 0 97.2%

   \[\leadsto -\frac{\color{blue}{\left(\log \left(\frac{4}{\pi}\right) - \log f\right)} \cdot 4}{\pi} \]

6. Final simplification 97.2%

   \[\leadsto \frac{4 \cdot \left(\log f - \log \left(\frac{4}{\pi}\right)\right)}{\pi} \]

Alternative 2: 96.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(-\frac{\log \left(f \cdot \frac{\pi}{4}\right)}{-\pi}\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* 4.0 (- (/ (log (* f (/ PI 4.0))) (- PI)))))

C

double code(double f) {
	return 4.0 * -(log((f * (((double) M_PI) / 4.0))) / -((double) M_PI));
}

Java

public static double code(double f) {
	return 4.0 * -(Math.log((f * (Math.PI / 4.0))) / -Math.PI);
}

Python

def code(f):
	return 4.0 * -(math.log((f * (math.pi / 4.0))) / -math.pi)

Julia

function code(f)
	return Float64(4.0 * Float64(-Float64(log(Float64(f * Float64(pi / 4.0))) / Float64(-pi))))
end

MATLAB

function tmp = code(f)
	tmp = 4.0 * -(log((f * (pi / 4.0))) / -pi);
end

Wolfram

code[f_] := N[(4.0 * (-N[(N[Log[N[(f * N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-Pi)), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 5.8%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

2. Taylor expanded in f around 0 97.2%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi}} \]
3. Step-by-step derivation

   1. *-commutative 97.2%

      \[\leadsto -\color{blue}{\frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} \cdot 4} \]

   2. associate-*l/ 97.2%

      \[\leadsto -\color{blue}{\frac{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f\right) \cdot 4}{\pi}} \]

   3. mul-1-neg 97.2%

      \[\leadsto -\frac{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + \color{blue}{\left(-\log f\right)}\right) \cdot 4}{\pi} \]

   4. unsub-neg 97.2%

      \[\leadsto -\frac{\color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) - \log f\right)} \cdot 4}{\pi} \]

   5. distribute-rgt-out-- 97.2%

      \[\leadsto -\frac{\left(\log \left(\frac{2}{\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}}\right) - \log f\right) \cdot 4}{\pi} \]

   6. metadata-eval 97.2%

      \[\leadsto -\frac{\left(\log \left(\frac{2}{\pi \cdot \color{blue}{0.5}}\right) - \log f\right) \cdot 4}{\pi} \]

4. Simplified 97.2%

   \[\leadsto -\color{blue}{\frac{\left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right) \cdot 4}{\pi}} \]

5. Taylor expanded in f around 0 97.2%

   \[\leadsto -\frac{\color{blue}{\left(\log \left(\frac{4}{\pi}\right) - \log f\right)} \cdot 4}{\pi} \]
6. Step-by-step derivation

   1. frac-2neg 97.2%

      \[\leadsto -\color{blue}{\frac{-\left(\log \left(\frac{4}{\pi}\right) - \log f\right) \cdot 4}{-\pi}} \]

   2. div-inv 97.1%

      \[\leadsto -\color{blue}{\left(-\left(\log \left(\frac{4}{\pi}\right) - \log f\right) \cdot 4\right) \cdot \frac{1}{-\pi}} \]

7. Applied egg-rr 97.1%

   \[\leadsto -\color{blue}{\left(4 \cdot \log \left(\frac{f}{\frac{4}{\pi}}\right)\right) \cdot \frac{1}{-\pi}} \]
8. Step-by-step derivation

   1. associate-*l* 97.1%

      \[\leadsto -\color{blue}{4 \cdot \left(\log \left(\frac{f}{\frac{4}{\pi}}\right) \cdot \frac{1}{-\pi}\right)} \]

   2. associate-*r/ 97.2%

      \[\leadsto -4 \cdot \color{blue}{\frac{\log \left(\frac{f}{\frac{4}{\pi}}\right) \cdot 1}{-\pi}} \]

   3. *-rgt-identity 97.2%

      \[\leadsto -4 \cdot \frac{\color{blue}{\log \left(\frac{f}{\frac{4}{\pi}}\right)}}{-\pi} \]

   4. associate-/r/ 97.2%

      \[\leadsto -4 \cdot \frac{\log \color{blue}{\left(\frac{f}{4} \cdot \pi\right)}}{-\pi} \]

   5. associate-*l/ 97.2%

      \[\leadsto -4 \cdot \frac{\log \color{blue}{\left(\frac{f \cdot \pi}{4}\right)}}{-\pi} \]

   6. associate-*r/ 97.2%

      \[\leadsto -4 \cdot \frac{\log \color{blue}{\left(f \cdot \frac{\pi}{4}\right)}}{-\pi} \]

9. Simplified 97.2%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(f \cdot \frac{\pi}{4}\right)}{-\pi}} \]

10. Final simplification 97.2%

    \[\leadsto 4 \cdot \left(-\frac{\log \left(f \cdot \frac{\pi}{4}\right)}{-\pi}\right) \]

Alternative 3: 95.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{\frac{4}{f}}{\pi}\right) \cdot \frac{-4}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (log (/ (/ 4.0 f) PI)) (/ (- 4.0) PI)))

C

double code(double f) {
	return log(((4.0 / f) / ((double) M_PI))) * (-4.0 / ((double) M_PI));
}

Java

public static double code(double f) {
	return Math.log(((4.0 / f) / Math.PI)) * (-4.0 / Math.PI);
}

Python

def code(f):
	return math.log(((4.0 / f) / math.pi)) * (-4.0 / math.pi)

Julia

function code(f)
	return Float64(log(Float64(Float64(4.0 / f) / pi)) * Float64(Float64(-4.0) / pi))
end

MATLAB

function tmp = code(f)
	tmp = log(((4.0 / f) / pi)) * (-4.0 / pi);
end

Wolfram

code[f_] := N[(N[Log[N[(N[(4.0 / f), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision] * N[((-4.0) / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.8%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

2. Taylor expanded in f around 0 97.2%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi}} \]
3. Step-by-step derivation

   1. *-commutative 97.2%

      \[\leadsto -\color{blue}{\frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} \cdot 4} \]

   2. associate-*l/ 97.2%

      \[\leadsto -\color{blue}{\frac{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f\right) \cdot 4}{\pi}} \]

   3. mul-1-neg 97.2%

      \[\leadsto -\frac{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + \color{blue}{\left(-\log f\right)}\right) \cdot 4}{\pi} \]

   4. unsub-neg 97.2%

      \[\leadsto -\frac{\color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) - \log f\right)} \cdot 4}{\pi} \]

   5. distribute-rgt-out-- 97.2%

      \[\leadsto -\frac{\left(\log \left(\frac{2}{\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}}\right) - \log f\right) \cdot 4}{\pi} \]

   6. metadata-eval 97.2%

      \[\leadsto -\frac{\left(\log \left(\frac{2}{\pi \cdot \color{blue}{0.5}}\right) - \log f\right) \cdot 4}{\pi} \]

4. Simplified 97.2%

   \[\leadsto -\color{blue}{\frac{\left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right) \cdot 4}{\pi}} \]

5. Taylor expanded in f around 0 97.2%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}} \]

7. Simplified 96.3%

   \[\leadsto -\color{blue}{\frac{4}{\pi} \cdot \log \left(\frac{\frac{4}{f}}{\pi}\right)} \]

8. Final simplification 96.3%

   \[\leadsto \log \left(\frac{\frac{4}{f}}{\pi}\right) \cdot \frac{-4}{\pi} \]

Alternative 4: 95.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(-\log \left(\frac{4}{\pi \cdot f}\right)\right)}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (* 4.0 (- (log (/ 4.0 (* PI f))))) PI))

C

double code(double f) {
	return (4.0 * -log((4.0 / (((double) M_PI) * f)))) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (4.0 * -Math.log((4.0 / (Math.PI * f)))) / Math.PI;
}

Python

def code(f):
	return (4.0 * -math.log((4.0 / (math.pi * f)))) / math.pi

Julia

function code(f)
	return Float64(Float64(4.0 * Float64(-log(Float64(4.0 / Float64(pi * f))))) / pi)
end

MATLAB

function tmp = code(f)
	tmp = (4.0 * -log((4.0 / (pi * f)))) / pi;
end

Wolfram

code[f_] := N[(N[(4.0 * (-N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 5.8%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

2. Taylor expanded in f around 0 97.2%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi}} \]
3. Step-by-step derivation

   1. *-commutative 97.2%

      \[\leadsto -\color{blue}{\frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} \cdot 4} \]

   2. associate-*l/ 97.2%

      \[\leadsto -\color{blue}{\frac{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f\right) \cdot 4}{\pi}} \]

   3. mul-1-neg 97.2%

      \[\leadsto -\frac{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + \color{blue}{\left(-\log f\right)}\right) \cdot 4}{\pi} \]

   4. unsub-neg 97.2%

      \[\leadsto -\frac{\color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) - \log f\right)} \cdot 4}{\pi} \]

   5. distribute-rgt-out-- 97.2%

      \[\leadsto -\frac{\left(\log \left(\frac{2}{\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}}\right) - \log f\right) \cdot 4}{\pi} \]

   6. metadata-eval 97.2%

      \[\leadsto -\frac{\left(\log \left(\frac{2}{\pi \cdot \color{blue}{0.5}}\right) - \log f\right) \cdot 4}{\pi} \]

4. Simplified 97.2%

   \[\leadsto -\color{blue}{\frac{\left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right) \cdot 4}{\pi}} \]
5. Step-by-step derivation

   1. expm1-log1p-u 95.8%

      \[\leadsto -\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right)\right)} \cdot 4}{\pi} \]

   2. diff-log 95.5%

      \[\leadsto -\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\log \left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)}\right)\right) \cdot 4}{\pi} \]

6. Applied egg-rr 95.5%

   \[\leadsto -\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)\right)\right)} \cdot 4}{\pi} \]
7. Step-by-step derivation

   1. expm1-log1p-u 96.8%

      \[\leadsto -\frac{\color{blue}{\log \left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)} \cdot 4}{\pi} \]

   2. associate-/l/ 96.8%

      \[\leadsto -\frac{\log \color{blue}{\left(\frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right)} \cdot 4}{\pi} \]

   3. *-commutative 96.8%

      \[\leadsto -\frac{\log \left(\frac{2}{f \cdot \color{blue}{\left(0.5 \cdot \pi\right)}}\right) \cdot 4}{\pi} \]

   4. associate-*r* 96.8%

      \[\leadsto -\frac{\log \left(\frac{2}{\color{blue}{\left(f \cdot 0.5\right) \cdot \pi}}\right) \cdot 4}{\pi} \]

8. Applied egg-rr 96.8%

   \[\leadsto -\frac{\color{blue}{\log \left(\frac{2}{\left(f \cdot 0.5\right) \cdot \pi}\right)} \cdot 4}{\pi} \]

9. Taylor expanded in f around 0 96.8%

   \[\leadsto -\frac{\log \color{blue}{\left(\frac{4}{f \cdot \pi}\right)} \cdot 4}{\pi} \]

10. Final simplification 96.8%

    \[\leadsto \frac{4 \cdot \left(-\log \left(\frac{4}{\pi \cdot f}\right)\right)}{\pi} \]

Alternative 5: 15.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \left|\frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}\right| \cdot \left(-4\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* (fabs (/ (log 7.62939453125e-6) PI)) (- 4.0)))

C

double code(double f) {
	return fabs((log(7.62939453125e-6) / ((double) M_PI))) * -4.0;
}

Java

public static double code(double f) {
	return Math.abs((Math.log(7.62939453125e-6) / Math.PI)) * -4.0;
}

Python

def code(f):
	return math.fabs((math.log(7.62939453125e-6) / math.pi)) * -4.0

Julia

function code(f)
	return Float64(abs(Float64(log(7.62939453125e-6) / pi)) * Float64(-4.0))
end

MATLAB

function tmp = code(f)
	tmp = abs((log(7.62939453125e-6) / pi)) * -4.0;
end

Wolfram

code[f_] := N[(N[Abs[N[(N[Log[7.62939453125e-6], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision] * (-4.0)), $MachinePrecision]

Derivation

1. Initial program 5.8%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

3. Applied egg-rr 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{262144}}\right) \]

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 0.0%

      \[\leadsto -4 \cdot \color{blue}{\left(\sqrt{\frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}} \cdot \sqrt{\frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}}\right)} \]

   2. sqrt-unprod 15.2%

      \[\leadsto -4 \cdot \color{blue}{\sqrt{\frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi} \cdot \frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}}} \]

   3. pow2 15.2%

      \[\leadsto -4 \cdot \sqrt{\color{blue}{{\left(\frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}\right)}^{2}}} \]

6. Applied egg-rr 15.2%

   \[\leadsto -4 \cdot \color{blue}{\sqrt{{\left(\frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}\right)}^{2}}} \]
7. Step-by-step derivation

   1. unpow2 15.2%

      \[\leadsto -4 \cdot \sqrt{\color{blue}{\frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi} \cdot \frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}}} \]

   2. rem-sqrt-square 15.2%

      \[\leadsto -4 \cdot \color{blue}{\left|\frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}\right|} \]

8. Simplified 15.2%

   \[\leadsto -4 \cdot \color{blue}{\left|\frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}\right|} \]

9. Final simplification 15.2%

   \[\leadsto \left|\frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}\right| \cdot \left(-4\right) \]

Alternative 6: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \log 0.5 \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (log 0.5) (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(0.5) * (-1.0 / (((double) M_PI) / 4.0));
}

Java

public static double code(double f) {
	return Math.log(0.5) * (-1.0 / (Math.PI / 4.0));
}

Python

def code(f):
	return math.log(0.5) * (-1.0 / (math.pi / 4.0))

Julia

function code(f)
	return Float64(log(0.5) * Float64(-1.0 / Float64(pi / 4.0)))
end

MATLAB

function tmp = code(f)
	tmp = log(0.5) * (-1.0 / (pi / 4.0));
end

Wolfram

code[f_] := N[(N[Log[0.5], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.8%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

3. Applied egg-rr 1.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{4}}\right) \]

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\log 0.5} \]

5. Final simplification 1.6%

   \[\leadsto \log 0.5 \cdot \frac{-1}{\frac{\pi}{4}} \]

Alternative 7: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi} \cdot \left(-4\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* (/ (log 7.62939453125e-6) PI) (- 4.0)))

C

double code(double f) {
	return (log(7.62939453125e-6) / ((double) M_PI)) * -4.0;
}

Java

public static double code(double f) {
	return (Math.log(7.62939453125e-6) / Math.PI) * -4.0;
}

Python

def code(f):
	return (math.log(7.62939453125e-6) / math.pi) * -4.0

Julia

function code(f)
	return Float64(Float64(log(7.62939453125e-6) / pi) * Float64(-4.0))
end

MATLAB

function tmp = code(f)
	tmp = (log(7.62939453125e-6) / pi) * -4.0;
end

Wolfram

code[f_] := N[(N[(N[Log[7.62939453125e-6], $MachinePrecision] / Pi), $MachinePrecision] * (-4.0)), $MachinePrecision]

Derivation

1. Initial program 5.8%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

3. Applied egg-rr 1.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{262144}}\right) \]

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi}} \]

5. Final simplification 1.6%

   \[\leadsto \frac{\log \left( 7.62939453125 \cdot 10^{-6} \right)}{\pi} \cdot \left(-4\right) \]

Reproduce

herbie shell --seed 2023285 
(FPCore (f)
  :name "VandenBroeck and Keller, Equation (20)"
  :precision binary64
  (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))) (- (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))))))))